1

Q0

A skydiver jumps off a plane at 3000m.

He falls with his belly “facing down” to 1000m altitude before opening up his parachute. Which is the qualitatively correct graph for velocity versus height?

Dynamics of a car/airplane/skytrain

and fuel economy

3

Work

• W = F d, where F is a force under consideration and d is the distance traveled.

• +/- signs: If the force is along the direction of motion, W is positive. If opposite, W is negative.

Net work is the sum of work done by different

forces or the work done by the net force.

4

Work - Energy Principle

• The net work on an object is zero if the net force is zero.

• A non-zero net work means non-zero net force. This is known as work-energy principle.

ifnet KKKW

5

Q1

When a person is pushing a crate on a rough surface with constant speed,

1) the work done by the person must be equal to the work done by friction forces.

2) the work done by the person must be greater to overcome frictions.

3) the work done by the person is less than the work done by friction forces.

6

Work-energy relation in transportation

W (net)=W (engine) – |W| (friction, Drag)= K

W (engine) =|W| (friction, Drag)+ K

A) During cruising at constant speed, K=0.

B) Acceleration costs extra energy!!

Energy in Transportation

All transportation systems need energy to

– Accelerate up to certain speed;– Make up for losses due to (rolling) friction, air

drag, etc. – Air planes need additional energy to climb to an

altitude of ~10 km.

Example

At 80-90 km/hr, the rolling friction and air drag of a car were estimated to be the same ( m= 1500 kg and r is given as 0.02 ).

Friction+ drag= 600 N.

Cars usually have their lowest energy consumption at ~ 80 to 90 km/h on an open road and consume 6 – 10 L of gasoline per 100 km depending on size and engine.

To maintain a constant speed,

W (engine)=|W| (frict, drag).

|W| = F d = 600 N x 100 km = 6 x 107 J.

Gasoline yields 34.6 MJ/per Litre and car engines’ efficiency is only ~ 20%.

This leads to a realistic answer of

8.5 L/100km.

Fuel economy of a car on a high way

Local versus highway

Gasoline consumption is larger in cities than on an open highway although speeds (or air drag forces) are lower. Why?

Extra energy for acceleration How much extra energy is consumed during acceleration?

To accelerate from 0 – 50 km/h in 5 s, the acceleration is a = 2.8 m/s2. The corresponding distance d = 35 m.

The rolling resistance and air drag combine to 400 N (following example 5.8). W = F d = 400 N x 35 m = 14000 J.

K = ½ m v2 = 150 000J (m = 1500 kg)A car needs at least 10 times more energy for acceleration.

12

Q1

In a previous lecture, we estimate that the drag force acting on a typical car is about 300N at speed 90km/hr. What is the drag force on this car when the speed is 135 km/hr?

A) 400N;

B) 450N;

C) 675N;

D) 900N.

Hybrid Cars

Hybrid cars more energy efficient because

• Efficiency of combustion engines depends on rpm; most efficient at relative high speeds ~ 90 km/h.

• Electric motor are very efficient at all speeds.

14

Power

• Power is defined as the rate at which work is done. Average power is the amount of work done in a time interval t or the amount of energy transferred:

SI unit: 1 watt = 1W = 1J/s

• Mechanical power:

t

E

t

WP

.||||

avgavg vFt

xF

t

WP

15

Fuel Economy of AirplanesA Boing 747 has a maximum range of 13,450 km and a maximum fuel capacity of 216,840 L.

We can calculate the fuel economy from this: 16L/km or 1600 L per 100 km.

The number of passengers is ~500, so each passenger uses 3.2L per 100 km when the plane is full.

This is similar to the fuel economy of a car!

16

The mass of the plane is 390 000 kg at take-off. V= m g h = 4 x 10^10 J

Kinetic energy at 915 km/h (254 m/s).

K = ½ m v^2 = 1.26 x 10^10 J

The energy efficiency of the jet engines is ~25% (estimated). Energy in jet fuel: 37.6 MJ/L.

So we need 1400 L/0.25 = 5600L of jet fuel to accelerate to 915 km/h and go up to 10500 m.

Energy needed to climb up to an altitude of 10.5km

17

Energy for cruising

• The rest of the fuel (~210,000L) is used to maintain “terminal” speed against air drag.

• Let’s estimate the drag force: A max. range of 13450km and 210,000 L are used

to counter air drag. So F = 147 kN. (Engines have max thrust of 4 x

282 kN). It is less than the air drag on the ground!!

Flying at a high altitude reduces the air drag because of a lower air density !! That’s why planes fly relatively high although going higher results in a lower upward lifting force.